Structure-preserving reduced order model for parametric cross-diffusion systems (Q6593976)

Cross-diffusion systems are often used to model the evolution of the concentrations or volumic fractions of mixtures composed of different species, and can also be used in population dynamics (as for instance in the SKT system). These systems often read as nonlinear degenerated parabolic partial differential equations, the numerical resolutions of which are highly expensive from a computational point of view. The goal of this work is to construct a structure-preserving Galerkin reduced-order model (ROM) for the resolution of parametric cross-diffusion systems. Indeed, the authors are interested in cross-diffusion systems which exhibit a so-called entropic structure, in the sense that they can be formally written as gradient flows of a certain entropy functional which is actually a Lyapunov functional of the system. This paper proposes a new reduced-order modelling method, based on a reduced basis paradigm, for the resolution of parameter-dependent cross-diffusion systems. This method preserves, at the ROM level, the main mathematical properties of the continuous solution, namely mass conservation, non-negativeness, preservation of the volume-filling property and entropy-entropy dissipation relationship. The theoretical advantages of this approach are illustrated by several numerical experiments.